Click this link or the diagram below to see a Flash animation of a typical turn signal system (you'll need to have Macromedia's Flash player installed). Click the left button for a left turn signal, the right button for a right turn, and the 4-way button for 4-way flashers. Click the Canel button to cancel the signals. Click the brake pedal with the mouse and hold the mouse button down to "apply" the brakes. Release the button to turn the brakes off. This will let you see how one signal can flash while the brake light comes on on the other side.
This will show you how one of these things works and should help if you are trying to troubleshoot your system. Many folks don't understand that their turn signal switch can cause problems like brake lights that don't work right. View the animation and you'll see why that's the case. The brake lights are routed right through the turn signal switch. The diagram below shows the contacts in position for a right turn with no brakes applied:
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